Answer Happy

Suppose your favorite professor has a collection of n different
Funko POP! figures that she needs to move from her office to her
home. She has a collection of unlabeled boxes to put the figures
in, and we will assume the boxes have a magical property that
allows any box to hold all n of the figures if she so chooses. (a)
How many ways are there to pack the figures into n − 1 boxes if
each box must have at least one figure contained within it? Explain
your reasoning. (b) How many ways are there to pack the figures
into n − 2 boxes if each box must have at least one figure
contained within it? Explain your reasoning. (c) How many ways are
there to pack the figures into 2 boxes if each box must have at
least one figure contained within it? Explain your reasoning. (d)
How many ways are there to pack the figures into 2 boxes if there
are no restrictions about how many figures can be contained within
a box? (That is, there could be an empty box.) Explain your
reasoning.